This fact is somewhat left implicit from the perspective of bipartite states \(\rho \in \mathcal {S}(\mathcal {A}\otimes \mathcal {B})\) . In order to make it explicit, we identify a bipartite state \(\rho \)  with its quantum channel \(\phi _\rho : \mathcal {A}\rightarrow \mathcal {B}\)  under the Choi-Jamiołkowski isomorphim [1]}, [2]} (see also, [3]}): recall that every quantum channel \(\phi : \mathcal {A}\rightarrow \mathcal {B}\) , i.e., every completely positive linear map, determines a bipartite state \(\rho _\phi \)  (up to normalisation) by
\(\rho _\phi = \sum _{ij} E_{ij} \otimes \phi (E_{ij})\; ,\footnote {Sometimes this is written as \rho _\phi = (\mathrm {id} \otimes \phi )(|\Phi \rangle \langle \Phi |), where |\Phi \rangle = \sum _i |i\rangle \otimes |i\rangle  is a maximally mixed state.}\) 
